A bigger dose doesn't get there faster

Take a pill on a schedule and the level in your blood climbs, dose after dose, toward a plateau. Double the dose and the plateau doubles — but it arrives at exactly the same moment. How high is up to the prescription. How long is up to the drug.

What you're seeing

Each dose lands as a vertical jump; between doses the body clears the drug, and the line decays. The catch is that the next dose arrives before the last one is gone, so it starts from a floor rather than from zero. That leftover is what builds. Each new peak is higher than the one before it — but by less each time, because a bigger level also means more drug cleared per hour. When the amount cleared between doses finally equals the amount of one dose, the climbing stops. That's the plateau, and the sawtooth just runs along it.

Now drag the dose. The whole picture stretches vertically: bigger jumps, a higher plateau. What doesn't move is the indigo curve — the share of the plateau reached so far — or the vertical line at five half-lives. That curve is 1 − e^−kt, and there is no dose in it and no interval in it. This is the part that surprises people: taking more does not get you to your steady level sooner. It only puts the steady level somewhere higher. Drag the interval slider and you'll see the same thing again — dosing twice as often raises the plateau, and the indigo curve still doesn't budge.

The only knob that moves the arrival time is the drug's own half-life. Drag it and watch the whole indigo curve and the five-half-lives marker slide. This is why a doctor starting you on a slow drug doesn't simply prescribe more of it — more would overshoot the target and still take a week. Instead they give a loading dose: one large first dose, sized exactly to fill the gap that accumulation would otherwise have taken days to fill. Press the button and the sawtooth starts at the plateau and stays there, from dose one.

The rule, exactly. A dose D into a volume V every τ hours, cleared first-order at k = ln2/t_½. Writing q = e^−kτ for the fraction still there when the next dose lands, the peak after n doses and the plateau it approaches are C_peak(n) = (D/V)·(1 − qⁿ)/(1 − q) ⟶ C_plateau = (D/V)/(1 − q) so the accumulation ratio is R = 1/(1−q) — exactly 2 when you dose once per half-life. Divide the first by the second and the dose cancels: the share of the plateau reached after time t is 1 − qⁿ = 1 − e^−kt = 1 − 2^−t/t_½ with no D and no τ anywhere in it — 96.9% after five half-lives, always. A loading dose of D·R lands on the plateau at dose one. Verified in node (improve/verify/80-drug-plateau.js, 22 checks): every closed form above is cross-examined against an independent fine-grained simulation of the actual dosing process — doses added, drug decayed step by step, formula never consulted — matching to 10⁻¹¹ mg/L across two very different regimes, and the loading-dose curve is confirmed to sit on the plateau and stay there. The measured time to 90% of plateau comes out identical for 50, 500 and 1000 mg, and halves when the half-life halves. Building this caught one of my own errors: the first version of that measurement reported 2.67 half-lives instead of 3.32, an off-by-one between "hours on the clock" and "intervals of therapy" — the closed form was right and my reading of it was not, which is precisely what cross-examining it is for. Four negative controls refute the beliefs this exhibit exists to kill: that a bigger dose arrives sooner (identical to 4 decimal places), that more frequent dosing arrives sooner (identical), that the level would plateau even with full washout between doses (a simulated no-carry-over model stays pinned at one dose forever), and that two half-lives is close enough (75%, a quarter short). One compartment, instant absorption, perfect adherence — real pharmacokinetics adds absorption time, tissue distribution and a missed dose on Thursday. The shape of the argument survives all three.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 80.